from __future__ import print_function
'''
Counting Summations
Problem 76
It is possible to write five as a sum in exactly six different ways:
4 + 1
3 + 2
3 + 1 + 1
2 + 2 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
How many different ways can one hundred be written as a sum of at least two positive integers?
'''
try:
xrange #Python 2
except NameError:
xrange = range #Python 3
def partition(m):
memo = [[0 for _ in xrange(m)] for _ in xrange(m+1)]
for i in xrange(m+1):
memo[i][0] = 1
for n in xrange(m+1):
for k in xrange(1, m):
memo[n][k] += memo[n][k-1]
if n > k:
memo[n][k] += memo[n-k-1][k]
return (memo[m][m-1] - 1)
print(partition(100))